The instantaneous center of a slider moving in a curved surface lies at the center of curvature.

Explanation:
To understand why the instantaneous center lies at the center of curvature, let's first define what the instantaneous center is. The instantaneous center is the point around which a body undergoes pure rotation, without any translational motion. It is the point where the velocity vectors of all points on the body are perpendicular to their respective position vectors.

When a slider is moving on a curved surface, it experiences both rotational and translational motion. The slider has a velocity vector associated with its translational motion, which is directed along the tangent to the curved surface at the point of contact between the slider and the surface. Additionally, the slider also has a rotational motion around some point.

To find the instantaneous center, we need to consider the relative velocities of two points on the slider. Let's consider two points - one on the slider and another on the curved surface, both in contact with each other. The velocity vector of the point on the slider is perpendicular to its position vector, as the slider is undergoing pure rotation. The velocity vector of the point on the curved surface is directed along the tangent to the curved surface at that point.

Since the slider is in contact with the curved surface, the two points have the same velocity magnitude and direction. This means that the velocity vector of the point on the curved surface is also perpendicular to its position vector. Therefore, the instantaneous center lies at the center of curvature, where the velocity vectors of both points are perpendicular to their respective position vectors.

Option C is correct because the instantaneous center of a slider moving in a curved surface lies at the center of curvature.